• <ins id="pjuwb"></ins>
    <blockquote id="pjuwb"><pre id="pjuwb"></pre></blockquote>
    <noscript id="pjuwb"></noscript>
          <sup id="pjuwb"><pre id="pjuwb"></pre></sup>
            <dd id="pjuwb"></dd>
            <abbr id="pjuwb"></abbr>

            O(1) 的小樂

            Job Hunting

            公告

            記錄我的生活和工作。。。
            <2025年7月>
            293012345
            6789101112
            13141516171819
            20212223242526
            272829303112
            3456789

            統計

            • 隨筆 - 182
            • 文章 - 1
            • 評論 - 41
            • 引用 - 0

            留言簿(10)

            隨筆分類(70)

            隨筆檔案(182)

            文章檔案(1)

            如影隨形

            搜索

            •  

            最新隨筆

            最新評論

            閱讀排行榜

            評論排行榜

            Kullback–Leibler divergence KL散度

            In probability theory and information theory, the Kullback–Leibler divergence[1][2][3] (also information divergence,information gain, relative entropy, or KLIC) is a non-symmetric measure of the difference between two probability distributions P and Q. KL measures the expected number of extra bits required to code samples from P when using a code based on Q, rather than using a code based on P. Typically P represents the "true" distribution of data, observations, or a precise calculated theoretical distribution. The measure Q typically represents a theory, model, description, or approximation of P.

            Although it is often intuited as a distance metric, the KL divergence is not a true metric – for example, the KL from P to Q is not necessarily the same as the KL from Q to P.

            KL divergence is a special case of a broader class of divergences called f-divergences. Originally introduced by Solomon Kullbackand Richard Leibler in 1951 as the directed divergence between two distributions, it is not the same as a divergence incalculus. However, the KL divergence can be derived from the Bregman divergence.

             

             

            注意P通常指數據集,我們已有的數據集,Q表示理論結果,所以KL divergence 的物理含義就是當用Q來編碼P中的采樣時,比用P來編碼P中的采用需要多用的位數!

             

            KL散度,也有人稱為KL距離,但是它并不是嚴格的距離概念,其不滿足三角不等式

             

            KL散度是不對稱的,當然,如果希望把它變對稱,

            Ds(p1, p2) = [D(p1, p2) + D(p2, p1)] / 2

             

            下面是KL散度的離散和連續定義!

            D_{\mathrm{KL}}(P\|Q) = \sum_i P(i) \log \frac{P(i)}{Q(i)}. \!

            D_{\mathrm{KL}}(P\|Q) = \int_{-\infty}^\infty p(x) \log \frac{p(x)}{q(x)} \; dx, \!

            注意的一點是p(x) 和q(x)分別是pq兩個隨機變量的PDF,D(P||Q)是一個數值,而不是一個函數,看下圖!

             

            注意:KL Area to be Integrated!

             

            File:KL-Gauss-Example.png

             

            KL 散度一個很強大的性質:

            The Kullback–Leibler divergence is always non-negative,

            D_{\mathrm{KL}}(P\|Q) \geq 0, \,

            a result known as , with DKL(P||Q) zero if and only if P = Q.

             

            計算KL散度的時候,注意問題是在稀疏數據集上KL散度計算通常會出現分母為零的情況!

             

             

            Matlab中的函數:KLDIV給出了兩個分布的KL散度

            Description

            KLDIV Kullback-Leibler or Jensen-Shannon divergence between two distributions.

            KLDIV(X,P1,P2) returns the Kullback-Leibler divergence between two distributions specified over the M variable values in vector X. P1 is a length-M vector of probabilities representing distribution 1, and P2 is a length-M vector of probabilities representing distribution 2. Thus, the probability of value X(i) is P1(i) for distribution 1 and P2(i) for distribution 2. The Kullback-Leibler divergence is given by:

               KL(P1(x),P2(x)) = sum[P1(x).log(P1(x)/P2(x))]

            If X contains duplicate values, there will be an warning message, and these values will be treated as distinct values. (I.e., the actual values do not enter into the computation, but the probabilities for the two duplicate values will be considered as probabilities corresponding to two unique values.) The elements of probability vectors P1 and P2 must each sum to 1 +/- .00001.

            A "log of zero" warning will be thrown for zero-valued probabilities. Handle this however you wish. Adding 'eps' or some other small value to all probabilities seems reasonable. (Renormalize if necessary.)

            KLDIV(X,P1,P2,'sym') returns a symmetric variant of the Kullback-Leibler divergence, given by [KL(P1,P2)+KL(P2,P1)]/2. See Johnson and Sinanovic (2001).

            KLDIV(X,P1,P2,'js') returns the Jensen-Shannon divergence, given by [KL(P1,Q)+KL(P2,Q)]/2, where Q = (P1+P2)/2. See the Wikipedia article for "Kullback–Leibler divergence". This is equal to 1/2 the so-called "Jeffrey divergence." See Rubner et al. (2000).

            EXAMPLE: Let the event set and probability sets be as follow:
               X = [1 2 3 3 4]';
               P1 = ones(5,1)/5;
               P2 = [0 0 .5 .2 .3]' + eps;
            Note that the event set here has duplicate values (two 3's). These will be treated as DISTINCT events by KLDIV. If you want these to be treated as the SAME event, you will need to collapse their probabilities together before running KLDIV. One way to do this is to use UNIQUE to find the set of unique events, and then iterate over that set, summing probabilities for each instance of each unique event. Here, we just leave the duplicate values to be treated independently (the default):
               KL = kldiv(X,P1,P2);
               KL =
                    19.4899

            Note also that we avoided the log-of-zero warning by adding 'eps' to all probability values in P2. We didn't need to renormalize because we're still within the sum-to-one tolerance.

            REFERENCES:
            1) Cover, T.M. and J.A. Thomas. "Elements of Information Theory," Wiley, 1991.
            2) Johnson, D.H. and S. Sinanovic. "Symmetrizing the Kullback-Leibler distance." IEEE Transactions on Information Theory (Submitted).
            3) Rubner, Y., Tomasi, C., and Guibas, L. J., 2000. "The Earth Mover's distance as a metric for image retrieval." International Journal of Computer Vision, 40(2): 99-121.
            4) <a href="
            http://en.wikipedia.org/wiki/Kullback%E2%80%93Leibler_divergence"&gt;Kullback–Leibler divergence</a>. Wikipedia, The Free Encyclopedia.

            posted on 2010-10-16 15:04 Sosi 閱讀(10020) 評論(2)  編輯 收藏 引用 所屬分類: Taps in Research

            評論

            # re: Kullback&ndash;Leibler divergence KL散度 2010-11-30 16:17 tintin0324

            博主,本人的研究方向需要了解kl距離,有些問題想請教下,怎么聯系呢?
              回復  更多評論    

            # re: Kullback&ndash;Leibler divergence KL散度 2010-12-05 22:37 Sosi

            @tintin0324
            KL 距離本身很簡單,如果就是那樣子定義的,意義也如上面所說。。如果你想深入了解的話,可以讀以下相關文獻
              回復  更多評論    
            統計系統
            亚洲午夜精品久久久久久人妖| 久久精品成人免费国产片小草| 九九热久久免费视频| 久久天天躁夜夜躁狠狠躁2022| 久久精品国产免费| 久久综合精品国产二区无码| 青青草国产97免久久费观看| 成人免费网站久久久| 亚洲色大成网站www久久九| 中文字幕无码av激情不卡久久| 久久精品无码av| 久久精品人人槡人妻人人玩AV | 国产免费久久精品丫丫| 亚洲国产精品成人久久| 久久久久久久久久久精品尤物 | 久久这里只有精品18| 久久久久久亚洲精品不卡| 久久A级毛片免费观看| 亚洲AV伊人久久青青草原| 亚洲国产二区三区久久| 欧洲人妻丰满av无码久久不卡| 久久久久久久亚洲精品| 久久97精品久久久久久久不卡| 亚洲国产精品久久久天堂| 精产国品久久一二三产区区别| 久久久WWW成人| 久久久久国产精品嫩草影院| 91精品国产综合久久久久久| 久久人人爽人人爽人人AV东京热| 亚洲欧美一区二区三区久久| 一本久久精品一区二区| 久久中文精品无码中文字幕| 少妇人妻综合久久中文字幕| 无码人妻精品一区二区三区久久| 亚洲国产婷婷香蕉久久久久久| 久久er国产精品免费观看8| 国产精品综合久久第一页 | 四虎国产精品免费久久5151| 亚洲国产精品久久久久婷婷软件 | 日韩精品无码久久久久久| 久久婷婷五月综合97色|